An important function f(x) = ex can be represented exactly by an infinite series called a power series. For example, the power series representation for ex is

For each real number x, it can be shown that the infinite series on the right converges to the number ex.

Power Series

a. The following power series is centered at 0.

b. The following power series is centered at –1.

c. The following power series is centered at 1.

Example 1 – Power Series

Radius and Interval of Convergence

Differentiation and Integration of Power Series

Taylor Series and Maclaurin Series